The Faber polynomials for \(m\)-fold symmetric domains
type: Item
instance of: scholarly article
title: The Faber polynomials for \(m\)-fold symmetric domains
zbMATH Open document ID: 0818.30002
DOI: 10.1016/0377-0427(94)90254-2
published in: Journal of Computational and Applied Mathematics
publication date: 1995-04-02T00:00:00Z
review text: Let \(E\) be a compact continuum in \(\mathbb{C}\), and let \(z= \psi(w)= w+ b_ 0+ {b_ 1\over w}+ {b_ 2\over w^ 2}+\cdots\) be the normalized conformal map from \(\{w: | w|> \rho\}\) onto the complement of \(E\). The Faber polynomials \(F_ n\) associated with \(E\) can be computed recursively if the \(b_ j\) are known. In the case that \(E\) is bounded by \(\{z\in \mathbb{C}: | z^ m- 1|= 1\}\) \((m= 2,3,\dots)\) or bounded by \(\{z\in \mathbb{C}: z= (w^ 2+ w+ 1)^{1/2}, | w|= 1\}\), the author gives explicit expressions for the \(F_ n\). Noteworthy is also the representation \(F_ n(z)= \text{det}(zI_ n- H_ n)\), where \(I_ n\) is the \(n\times n\) unit matrix, and where \(H_ n\) is an \(n\times n\) matrix in which the \(b_ j\) enter. The zeros of \(F_ n\) are therefore the eigenvalues of \(H_ n\). In the examples studied, these zeros are dense on the lemniscates.
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 739562
zbMATH Keywords: Faber polynomials
zbMATH Keywords: zeros of polynomials
author: Matthew He
reviewed by: Dieter Gaier
MaRDI profile type: Publication
full work available at URL: 0377-0427(94)90254-2
OpenAlex ID: W2070276434
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description: scientific article; zbMATH DE number 739562